Moderate -0.3 This is a standard M2 momentum and collisions question requiring straightforward application of conservation of momentum and coefficient of restitution formulae. Part (a) involves routine algebraic manipulation with given expressions to show, while part (b) applies impulse-momentum and 2D collision principles. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average for A-level.
Two small objects, P of mass \(m \mathrm {~kg}\) and Q of mass \(k m \mathrm {~kg}\), slide on a smooth horizontal plane. Initially, P and Q are moving in the same straight line towards one another, each with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
After a direct collision with P , the direction of motion of Q is reversed and it now has a speed of \(\frac { 1 } { 3 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of P is now \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where the positive direction is the original direction of motion of P .
Draw a diagram showing the velocities of P and Q before and after the impact.
By considering the linear momentum of the objects before and after the collision, show that \(v = \left( 1 - \frac { 4 } { 3 } k \right) u\).
Hence find the condition on \(k\) for the direction of motion of P to be reversed.
The coefficient of restitution in the collision is 0.5 .
Show that \(v = - \frac { 2 } { 3 } u\) and calculate the value of \(k\).
Particle \(A\) has a mass of 5 kg and velocity \(\binom { 3 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Particle \(B\) has mass 3 kg and is initially at rest. A force \(\binom { 1 } { - 2 } \mathrm {~N}\) acts for 9 seconds on B and subsequently (in the absence of the force), \(A\) and \(B\) collide and stick together to form an object \(C\) that moves off with a velocity \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\).
Show that \(\mathbf { V } = \binom { 3 } { - 1 }\).
The object C now collides with a smooth barrier which lies in the direction \(\binom { 0 } { 1 }\). The coefficient of restitution in the collision is 0.5 .
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\begin{enumerate}[label=(\alph*)]
\item Two small objects, P of mass $m \mathrm {~kg}$ and Q of mass $k m \mathrm {~kg}$, slide on a smooth horizontal plane. Initially, P and Q are moving in the same straight line towards one another, each with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
After a direct collision with P , the direction of motion of Q is reversed and it now has a speed of $\frac { 1 } { 3 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The velocity of P is now $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where the positive direction is the original direction of motion of P .
\begin{enumerate}[label=(\roman*)]
\item Draw a diagram showing the velocities of P and Q before and after the impact.
\item By considering the linear momentum of the objects before and after the collision, show that $v = \left( 1 - \frac { 4 } { 3 } k \right) u$.
\item Hence find the condition on $k$ for the direction of motion of P to be reversed.
The coefficient of restitution in the collision is 0.5 .
\item Show that $v = - \frac { 2 } { 3 } u$ and calculate the value of $k$.
\end{enumerate}\item Particle $A$ has a mass of 5 kg and velocity $\binom { 3 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Particle $B$ has mass 3 kg and is initially at rest. A force $\binom { 1 } { - 2 } \mathrm {~N}$ acts for 9 seconds on B and subsequently (in the absence of the force), $A$ and $B$ collide and stick together to form an object $C$ that moves off with a velocity $\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\mathbf { V } = \binom { 3 } { - 1 }$.
The object C now collides with a smooth barrier which lies in the direction $\binom { 0 } { 1 }$. The coefficient of restitution in the collision is 0.5 .
\item Calculate the velocity of C after the impact.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI M2 2009 Q1 [18]}}